Almost soft compact and approximately soft Lindelöf spaces

ABSTRACT The concepts of almost soft compact (almost soft Lindelöf), approximately soft Lindelöf and mildly soft compact (mildly soft Lindelöf) spaces are initiated and investigated in this work. Their characterizations and main properties are established and the relationships among them are illustrated with the help of examples. The sufficient conditions for the equivalent between almost soft Lindelöf and approximately soft Lindelöf spaces; and for the equivalent among soft Lindelöf, almost soft Lindelöf, approximately soft Lindelöf and mildly soft Lindelöf spaces are given. The image of the introduced types of soft compact and soft Lindelöf spaces under soft continuous mappings are studied; and the mutual relations between them and their parametric topological spaces are probed. Some results which associate soft hyperconnectedness and soft connectedness, respectively with almost soft compactness and mildly soft compactness are verified. This study is concluded by providing an illustrative diagram for the interrelations among the initiated soft compact and soft Lindelöf spaces.


Introduction
Many practical problems in physics, engineering, medical and social science and economic suffer from data that involve uncertainties. For this reason, Molodtsov [1] proposed soft sets as a new mathematical tool to deal with uncertainties. He showed the advantages of soft sets compared to fuzzy sets and he successfully applied soft sets in several directions such as game theory and operations research. Maji et al. [2,3] gave an application of soft sets and initiated some operators on soft sets in 2002 and 2003, respectively. Later on, many researches related to the soft set theory and its applications appeared rapidly (see, for example, [4][5][6][7][8]). A novel idea of soft topological spaces was introduced in 2011 by Shabir and Naz [9]. They presented the notions of soft open and soft closed sets, soft neighbourhoods and soft separation axioms; and investigated their fundamental features. In 2012, Aygünoǧlu and Aygün [10] introduced a concept of soft compact spaces and derived main properties. They also presented a notion of enriched soft topological spaces and illuminated its role to verified some results related to constant soft mappings and soft compact spaces. Zorlutuna et al. [11] came up with an idea of soft points and employed it to study some properties of soft interior points and soft neighbourhood systems. In [12,13], the authors simultaneously modified a concept of soft point for studying soft metric spaces and soft neighbourhood systems. Karaaslan [14], in 2016, defined the concepts of lower and upper approximations of a soft set based on soft class operations introduced therein. He ended his study by constructing a decision making method utilizing soft rough class; and providing an example to demonstrate that the suggested method can successfully work. In [15][16][17], the authors carried out three studies to correct some alleged results via soft topologies. Recently, we [18] introduced the notions of partial belong and total non belong relations and investigated main properties. Also, we [19] initiated a concept of soft topological ordered spaces and introduced the concepts of monotone soft sets and p-soft T i -ordered spaces (i = 0, 1,2,3,4).
In this article, we first formulate the concepts of almost soft compact and almost soft Lindelöf spaces; and deduce their main properties. Also, we verify that (X, τ , K) is an enriched almost soft compact space if and only if (X, τ k ) is almost compact, for each k ∈ K. Second, we define approximately soft Lindelöf spaces and point out their equivalent with almost soft Lindelöf spaces if the space is soft locally finite. Also, we give an answer of why do not define approximately soft compact spaces. Finally, we present the notions of mildly soft compact and mildly soft Lindelöf spaces; and derive their basic properties. One of important results, in the last section, is Theorem 5.14, which illustrates the equivalent among soft Lindelöf, almost soft Lindelöff, approximately soft Lindelöf and mildly soft Lindelöf spaces if the space is soft partition. In general, we give a completely description for each type of the soft compact and soft Lindelöf spaces introduced herein; and conclude that all of them are preserved under soft continuous mappings. We proved some results related to enriched soft topological spaces; and we explore some results which associate soft hyperconnected and soft connected spaces, respectively with almost soft compact and mildly soft compact spaces.

Preliminaries
In this section, we mention some definitions and preliminaries results which will be needed in the sequels.

Definition 2.1 ([1]):
A pair (G, K) is called a soft set over a non-empty set X provided that G is a mapping of a set of parameters K into the family of all subsets of X. It can be written as follows:

Definition 2.3 ([5]):
The relative complement of a soft set (G, K), denoted by (G, K) c , is given by We draw the attention of the readers to the existence of another definition of complement of a soft set given in [3]. With regard to this definition, the De Morgan's laws do not keep via the soft set theory.

Definition 2.4 ([5,20]):
The soft union and intersection of two soft sets (G, K), (F, K) are defined, respectively, as follows: It worthily noting that the soft union and intersection of an arbitrary family of soft sets were given in [21,22], respectively.
A collection of all soft subsets of X is denoted by S(X K ). Definition 2.6 ([12,13]): A soft set (P, K) over X is called soft point if there is k ∈ K and there is x ∈ X satisfies that P(k) = {x} and P(e) = ∅, for each e ∈ K \ {k}.
A soft point will be shortly denoted by P x k .

Definition 2.8 ([9]):
For a soft set (G, K) over X and x ∈ X, we say that x ∈ (G, K) if x ∈ G(k), for each k ∈ K; and we say that Proposition 2.13 ([9]): Let (L, K) be a soft subset of (X, τ , K). Then: In the literature, there are many different notions of a soft Hausdorff space. In this study, we investigate two types of them and distinguish between them by writing T 2 and T 2 . They are defined as follows: Definition 2.14: An STS (X, τ , K) is said to be: [9] if for every x = y ∈ X, there are two disjoint soft open sets (G, K) and (F, K) such that x ∈ (G, K) and y ∈ (F, K). (ii) Soft T 2 -space [24] if for every two soft points P x k , P y e such that x = y, there are two disjoint soft open sets (G, K) and (F, K) such that P x k ∈ (G, K) and P y e ∈ (F, K).

Definition 2.16 ([25]):
A soft set (F, K) over X is said to be pseudo constant provided that F(k) = X or ∅, for each k ∈ K.
A family of all pseudo constant soft sets is briefly denoted by CS(X, K).

Definition 2.19 ([10]):
A soft topology τ on X is said to be enriched if a condition (i) of Definition 2.9 is replaced by the following condition: (G, K) ∈ τ , for all (G, K) ∈ CS(X, K). In this case, we term the triple (X, τ , K) an enriched STS.  By analogy with the above definition and theorem, we introduce the following definition and result.

Proposition 2.29 ([13]): Let f φ : S(X A ) → S(Y B ) be a soft mapping. Then for each soft subsets G A and H B of S(X A )
and S(Y B ), respectively, we have the following results:  (i) f φ is soft continuous; (ii) The inverse image of each soft open (resp. soft closed) set is soft open (resp. soft closed); Throughout this paper, we utilize the following notations:

Definition 2.32 ([18]): A soft set
S to indicate to a countable set. (iii) R and N to indicate to the set of real numbers and the set of natural numbers, respectively.

Almost soft compact and almost soft Lindelöf spaces
In this section, the notions of almost soft compact and almost soft Lindelöf spaces are formulated and several properties of them are given. Some interesting results which associate almost soft compact spaces with the two concepts of enriched STSs and soft T 2 -spaces are presented and discussed.
Definition 3.1: An STS (X, τ , K) is called almost soft compact (resp. almost soft Lindelöf) if every soft open cover of X has a finite (resp. countable) soft sub-cover the soft closure of whose members cover X.
For the sake of economy, the proofs of the following four propositions will be omitted.
In general, the converse of Propositions 3.2, 3.3 and 3.5 is not true as evidenced by the two examples below. Example 3.6: Consider (N , τ , K) is a soft topological space such that K = {k 1 , k 2 } and τ is the discrete soft topology. Obviously, (N , τ , K) is soft Lindelöf but it is not almost soft compact.
A similar proof is given in the case of an almost soft Lindelöf space.

Definition 3.11:
It is clear that a collection satisfies the first type of finite (resp. countable) intersection property, it also satisfies the finite (resp. countable) intersection property.
Theorem 3.12: An STS (X, τ , K) is almost soft compact (resp. almost soft Lindelöf) if and only if every collection of soft closed subsets of (X, τ , K), satisfying the first type of finite (resp. countable) intersection property, has, itself, a non-null soft intersection.
Proof: We will start with the proof for almost soft compactness, because the proof for almost soft Lindelöfness is analogous.
Hence the necessary condition holds.
Conversely, let be a collection of soft closed subsets of X which satisfies the first type of finite intersection property. Then it also satisfies the finite intersection property. Since has a non-null soft intersection, then (X, τ , K) is a soft compact space. It follows, by Proposition 3.5, that (X, τ , K) is almost soft compact.

Proposition 3.13:
The soft continuous image of an almost soft compact (resp. almost soft Lindelöf) set is almost soft compact (resp. almost soft Lindelöf).
is almost soft Lindelöf, as desired.
A similar proof is given in the case of an almost soft compact space. In the following, we construct an example to show that the converse of the above proposition is not true in general.  This implies that x ∈ H(k). Thus P x k ∈ (H, K). Hence (H, K) ⊆(H, K). From Proposition 2.13, we obtain (H, K) = (H, K). Hence the desired result is proved.

Remark 3.18:
It can be seen that an enriched almost soft Lindelöf (resp. enriched almost soft compact) space (X, τ , K) implies that a set of parameters K is countable (resp. finite).
Proof: To prove the proposition in the case of (X, τ k ) is almost Lindelöf, let {(G j , K) : j ∈ J} be a soft open cover of (X, τ , K) such that K is countable. Say, | E |= ℵ, where ℵ is the cardinal number of the natural numbers set. Then X = j∈J G j (k) for each k ∈ K. As (X, τ k ) is almost Lindelöf for each k ∈ K, then there exist countable sets M n such that X = j∈M 1 G j (k 1 ), X = Proof: We prove the theorem in the case of an enriched almost soft compact space and the case between parenthesis is made similarly.
[⇒] Let {H j (k) : j ∈ J} be an open cover for (X, τ k ). By hypothesis, (X, τ , K) is enriched, so we can construct a soft open cover of (X, τ , K) consisting of the following soft sets: Obviously, {(F j , K) (G, K) : j ∈ J} is a soft open cover of (X, τ , K). As (X, τ , K) is almost soft compact, then Hence (X, τ k ) is almost compact.
[⇐] The proof of the sufficient part comes immediately from Remark 3.18 and Proposition 3.19. Consider ((A, K), τ (A,K) , K) is a soft subspace of (X, τ , K) and let (G, K) A and (G, K) •A stand for the soft closure and soft interior operators, respectively, in ((A, K), τ (A,K) , K). Then:

Lemma 3.21:
is an almost soft Lindelöf subset of (X, τ , K).   ∈ (A, K). So by using a similar technique of the above proof, the corollary holds. Proof: It is enough to prove that g φ is soft closed.

Proof: Let the given conditions be satisfied and let
Obviously, every almost soft compact partition space (X, τ , K) is soft compact. Let (A, K) be a soft closed subset of (X, τ , K).
is soft closed. Hence g φ is soft homeomorphism.

Proposition 3.27:
If there exists a finite soft dense subset of (X, τ , K) such that K is finite, then (X, τ , K) is almost soft compact.
a soft open cover of (X, τ , K) and let (D, K) be a soft dense subset of (X, τ , K). K is a finite set and (D, K) is a finite soft set, then a collection {(G x s , K)} is finite. Thus X = (G x s , K). Hence the proof is complete.

Approximately soft Lindelöf spaces
We define in this section an approximately soft Lindelöf spaces concept which is wider than an almost soft Lindelöf spaces concept and illustrate under what conditions almost soft Lindelöf and approximately soft Lindelöf spaces are equivalent. Also, we study several properties concerning approximately soft Lindelöf and soft locally finite spaces.

Remark 4.2:
If we replace a word "countable" in the above definition by "finite", then we obtain a definition of an almost soft compact space because K). For this reason, we do not define approximately soft compact spaces.

Proposition 4.3: Every almost soft Lindelöf space is approximately soft Lindelöf.
Proof: Since i∈I (G i , K) ⊆ i∈I (G i , K), then the proposition is satisfied. The next example shows that the converse of Proposition 4.3 is not true in general.

Example 4.5:
Let {V i : i ∈ I} be a collection of all open subsets of the usual topological space (R, U) and let K = {k 1 , k 2 } be a set of parameters. We construct a soft topology τ on R as follows: a soft set (H, K) which is given by H(k 1 ) = H(k 2 ) = Q is countable soft dense, then it follows from Proposition 4.7, that (R, τ , K) is approximately soft Lindelöf. On the other hand, a soft a soft open cover of R. Since this collection has not a countable soft sub-collection such that the soft closure of whose members cover R, then (R, τ , K) is not almost soft Lindelöf.  (G i , K), K). Hence the desired result is proved.

Proposition 4.7:
If there exists a countable soft dense subset of (X, τ , K) such that K is countable, then (X, τ , K) is approximately soft Lindelöf.

Proof:
The proof is similar of that Proposition 3.27.  (H i , K).  It is clear that any collection satisfies the second type of countable intersection property, it also satisfies the first type of countable intersection property. An STS (X, τ , K) is approximately soft Lindelöf if and only if every collection of soft closed subsets of (X, τ , K), satisfying the second type of countable intersection property, has, itself, a non-null soft intersection. K)) • , as required. Sufficiency: Let be a collection of soft closed subsets of X which satisfies the second type of countable intersection property. Then it also satisfies the first type of countable intersection property. Since has a nonnull soft intersection, then (X, τ , K) is an almost soft Lindelöf space. It follows, by Proposition 4.3, that (X, τ , K) is approximately soft Lindelöf.

Theorem 4.21:
If a collection {(G i , K) : i ∈ I} of (X, τ , K) is soft locally finite, then i∈I (G i , K) = i∈I (G i , K).
Conversely, let P x k ∈ i∈I (G i , K). Then we can find a finite set M ⊆ I satisfies that P x k ∈ (G m , K), for each m ∈ M.
Therefore K). This completes the proof.  To prove the sufficient condition, let (X, τ , K) be approximately soft Lindelöf and consider {(G i , K) : i ∈ I} is a soft open cover of X. By hypothesis, X ⊆ s∈S (G i , K).
As (X, τ , K) is a soft locally finite space, then s∈S (G i , K) = s∈S (G i , K). Hence the desired result is proved.

Mildly soft compact and mildly soft Lindelöf spaces
The concepts of mildly soft compact and mildly soft Lindelöf spaces are presented and their basic features are studied. The necessary and sufficient conditions for them are given. A sufficient condition for the equivalent among soft Lindelöf, almost soft Lindelöf, approximately soft Lindelöf and mildly soft Lindelöf spaces is investigated.
Definition 5.1: An STS (X, τ , K) is called mildly soft compact (resp. mildly soft Lindelöf) if every soft clopen cover of X has a finite (resp. countable) soft sub-cover.
The proofs of the following two propositions are easy and thus omitted.
By the next example, we illuminate that the converse of the above proposition is not true.
Example 5.7: Assume that (R, τ , K) is the same as in Example 4.5. We show that (R, τ , K) is not almost soft Lindelöf. It can be observed that the only soft clopen subsets of (R, τ , K) are R and ∅. So (R, τ , K) is mildly soft compact.
For the sake of economy, the proofs of the following three results will be omitted. Proof: In view of (X, τ , K) is soft connected, then the only soft clopen subsets of (X, τ , K) are X and ∅. Hence (X, τ , K) is mildly soft compact.
The next example illustrates that the converse of the above proposition fails.  (iii) → (iv): Let {(G i , K) : i ∈ I} be a clopen cover of X. As (X, τ , K) is approximately soft Lindelöf, then X ⊆ s∈S (G i , K) and as (X, τ , K) is soft partition, then be a soft open cover of X. As (X, τ , K) is soft partition, then {(G i , K) : i ∈ I} is a clopen cover of X and as (X, τ , K) is mildly soft Lindelöf, then X = s∈S (G i , K). This completes the proof.

Proof:
The necessary condition is obvious.
To verify the sufficient condition, assume that is a soft open cover of a mildly soft compact space (X, τ , K). Since X is a soft union of members of the soft base and X is mildly soft compact, then we can find a finite member (H s , K) of the soft base satisfies that X =  Necessity: Let {H j (k) : j ∈ J} be a clopen cover for (X, τ k ). By the above lemma, we can construct a soft clopen cover of (X, τ , K) consisting of the following soft sets, all soft clopen sets (F j , K) in which F j (k) = H j (k) and F j (k i ) = X, for each k i = k. Obviously, {(F j , K) : j ∈ J} is a soft clopen cover of (X, τ , K). As (X, τ , K) is mildly soft Lindelöf, then X = j∈S (F j , K). Thus X = j∈S F j (k) = j∈S H j (k). Hence (X, τ k ) is mildly Lindelöf. Sufficiency: Let {(G j , K) : j ∈ J} be a soft clopen cover of (X, τ , K). Since (X, τ , K) is enriched, then K is countable. Say, | E |= ℵ, where ℵ is the cardinal number of the natural numbers set. Then X = j∈J G j (k) for each k ∈ K. As (X, τ k ) is mildly Lindelöf for each k ∈ K, then there exist countable sets M n such that X = j∈M 1 G j (k 1 ), X = j∈M 2 G j (k 2 ), . . . , X = j∈M m G j (k m ), . . . . Therefore X = j∈ n∈ℵ M n (G j , K). Thus (X, τ , K) is mildly soft Lindelöf.
In the following, the proofs of Proposition 5.19 and Proposition 5.20 are similar to the proofs of Proposition 3.19 and Proposition 3.22, respectively.

Proposition 5.19:
If (X, τ k ) is mildly compact (resp. mildly Lindelöf), for each k ∈ K such that K is finite (resp. countable), then (X, τ , K) is mildly soft compact (resp. mildly soft Lindelöf). We demonstrate in the next example that an approximately soft Lindelöf need not be a mildly soft Lindelöf space.

Conclusion
In this work, the concepts of almost soft compact (almost soft Lindelöf), approximately soft Lindelöf and mildly soft compact (mildly soft Lindelöf) spaces are introduced and studied. The relationships among these introduced concepts are illustrated and their relationships with some soft topological notions such as soft connected and soft hyperconnected spaces are shown with the help of examples. The notions of soft locally finite and soft partition spaces are presented and then they are used to verify some important results such as Theorem 4.23 and Theorem 5.14. As well as, the given concepts of the first type of finite (countable) intersection property and the second type of countable intersection property are utilized to characterize almost soft compact (almost soft Lindelöf) and approximately soft Lindelöf spaces, respectively. Some findings concerning soft subspaces and enriched soft topological spaces are investigated in detail. The presented concepts in this study are elementary and fundamental for further researches and will open a way to improve more applications on soft topology.